0,3

a(n) = degree of Kac determinant at level n as polynomial in the conformal weight (called h). (Cf. C. Itzykson and J.-M. Drouffe, Statistical Field Theory, Vol. 2, p. 533, eq.(98); reference p. 643, Cambridge University Press, (1989)) - comment from Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)

Also the number of one-element transitions from the integer partitions of n to the partitions of n-1 for labeled parts with the assumption that from any part z > 1 one can take an element of amount 1 in one way only. That means z is composed of z unlabeled parts of amount 1, i.e. z = 1 + 1 + ... + 1. E.g. for n=3 to n=2 we have A006218(3) = 6 and [111] --> [11], [111] --> [11], [111] --> [11], [12] --> [11], [12] --> [2], [3] --> [2]. For the case of z composed by labeled elements, z = 1_1 + 1_2 + ... + 1_z, refer to A066186. - Thomas Wieder (wieder.thomas(AT)t-online.de), May 20 2004

Number of times a derivative of any order (not 0 of course) appears when expanding the n-th derivative of 1/f(x). For instance (1/f(x))'' = (2 f'(x)^2-f(x) f''(x)) / f(x)^3 which makes a(2) = 3 (by counting k times the k-th power of a derivative). - Thomas Baruchel (baruchel(AT)users.sourceforge.net), Nov 07 2005

Starting with offset 1, = the partition triangle A008284 * [1, 2, 3,...]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Feb 13 2008

Starting with offset 1 equals A000041: (1, 1, 2, 3, 5, 7, 11,...) convolved with A000005: (1, 2, 2, 3, 2, 4,...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 16 2009]

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Erdos, Paul and Lehner, Joseph, The distribution of the number of summands in the partitions of a positive integer, Duke Math. J. 8, (1941). 335-345.

Kessler, I. and Livingston, M., The expected number of parts in a partition of n, Monatsh. Math. 81 (1976), no. 3, 203-212.

T. D. Noe, Table of n, a(n) for n=0..1000

G.f.: Sum n*x^n Product 1/(1-x^k); k = 1..n; n=1..inf.

G.f.: Sum x^k/(1-x^k); k=1..inf * Product 1/(1-x^m); m=1..inf.

a(n) = Sum of the number of divisors of m * partitions of n-m; m=1..n.

g:=sum(n*x^n*product(1/(1-x^k), k=1..n), n=1..60);

f[n_] := Sum[DivisorSigma[0, m] PartitionsP[n - m], {m, 1, n}]; Table[ f[n], {n, 0, 41}]

CoefficientList[ Series[ Sum[n*x^n*Product[1/(1 - x^k), {k, n}], {n, 100}], {x, 0, 100}], x]

(PARI) f(n)= {local(v, i, k, s, t); v=vector(n, k, 0); v[n]=2; t=0; while(v[1]<n, i=2; while(v[i]==0, i++); v[i]--; s=sum(k=i, n, k*v[k]); while(i>1, i--; s+=i*(v[i]=(n-s)\i)); t+=sum(k=1, n, v[k])); t } (Thomas Baruchel (baruchel(AT)users(AT)sourceforge.net), Nov 07 2005)

a(n) = sum(k*A008284(n, k), k=1..n).

Cf. A093694, A066186, A000070.

Cf. A008284.

A000041, A000005 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 16 2009]

Sequence in context: A001975 A096220 A034333 this_sequence A079983 A028926 A038577

Adjacent sequences: A006125 A006126 A006127 this_sequence A006129 A006130 A006131

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com), clm